Step 1

Place a rack in middle of oven and preheat to 350°. Toast 2½ cups (350 g) hazelnuts, preferably blanched, on a rimmed baking sheet until golden brown in the center, 10–12 minutes. Let cool. If nuts are unblanched, rub together in a clean kitchen towel to remove skins; discard skins.

Step 2

Using a paper towel, lightly grease bottom and sides of tart pan with vegetable oil. If you’d like to eventually remove tart from bottom of pan, line with a parchment paper round. Melt 7 oz. semisweet chocolate (about 60% cacao), chopped (about 1 cup), 7 oz. milk chocolate, chopped (about 1 cup), and ¼ tsp. Diamond Crystal or Morton kosher salt in a heatproof bowl set over a saucepan filled with 1" of barely simmering water, stirring with a heatproof rubber spatula, until just melted and smooth, about 5 minutes. Pour about half of chocolate into prepared pan (eyeballing is fine!); set remaining chocolate aside. Using a small spoon, spread chocolate across bottom and up sides of pan, working it all the way to edges of pan. Place on a rimmed baking sheet or large plate and chill in freezer until chocolate is set, at least 10 minutes and up to 30 minutes.

Step 3

Meanwhile, pulse hazelnuts and remaining 2 tsp. Diamond Crystal or 1¼ tsp. Morton kosher salt in a food processor until a paste forms, about 4 minutes. Add 4 Tbsp. unsalted butter, room temperature, 1 Tbsp. vanilla bean paste or vanilla extract, and 1 Tbsp. vegetable oil; process until the texture of smooth peanut butter, about 4 minutes. Add 1¼ cups (138 g) powdered sugar; pulse until combined (mixture will be thick).

Step 4

Crush 4 sugar or waffle ice cream cones in a medium bowl until pieces are about peanut-size. Scrape hazelnut butter into bowl and mix until well combined (using your hands might be most efficient). Scrape hazelnut butter filling into chocolate shell and gently press down into shell to compact into an even layer; smooth surface with an offset spatula.

Step 5

Scrape remaining melted chocolate over nut butter (reheat slightly if needed), spreading all the way to the edges. Smooth surface with offset spatula or spoon. Sprinkle generously with flaky sea salt, then chill tart in freezer until chocolate is set and sides of tart are pulling away from pan, at least 30 minutes and up to 4 hours.

Step 6

Gently pop tart out of pan. (If pan is lined with parchment paper, remove bottom and peel away parchment; discard.) Transfer tart to a large plate and thaw in fridge 30 minutes.

Step 7

Let tart sit at room temperature 15–20 minutes before cutting into thin wedges.Do ahead: Tart can be made 2 days ahead. Cover loosely with plastic wrap and keep chilled.